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(C) Case $(I)$: The system consists of two masses $m$ and $2m$ connected by a string over a pulley. The equation of motion for mass $m$ is $T - mg = m

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less than that in case $II$

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(C) Case $(I)$: The system consists of two masses $m$ and $2m$ connected by a string over a pulley. The equation of motion for mass $m$ is $T - mg = ma$, and for mass $2m$ is $2mg - T = 2ma$. Adding these equations gives $mg = 3ma$, which results in $a_I = g/3$.Case $(II)$: The mass $m$ is pulled by a constant force $F = 2mg$. The equation of motion for mass $m$ is $F - mg = ma$. Substituting $F = 2mg$, we get $2mg - mg = ma$, which simplifies to $mg = ma$, resulting in $a_{II} = g$.Comparing the accelerations, $a_I = g/3$ and $a_{II} = g$. Therefore, the acceleration in case $I$ is less than that in case $II$.

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